Previous research has shown that cache behavior is important for main memory index structures. Cache conscious index structures such as Cache Sensitive Search Trees (CSS-Trees) perform lookups much faster than binary search and T-Trees. However, CSS-Trees are designed for decision support workloads with relatively static data. Although B +-Trees are more cache conscious than binary search and T-Trees, their utilization ofacachelineislowsincehalfofthespaceisused to store child pointers. Nevertheless, for applications that require incremental updates, traditional B +-Trees perform well. Our goal is to make B +-Trees as cache conscious as CSS-Trees without increasing their update cost too much. We propose a new indexing technique called “Cache Sensitive B +-Trees ” (CSB +-Trees). It is a variant of B +-Trees that stores all the child nodes of any given node contiguously, and keeps only the address of the first child in each node. The rest of the children can be found by adding an offset to that address. Since only one child pointer is stored explicitly, the utilization of a cache line is high. CSB +-Trees support incremental updates in a way similar to B +-Trees. We also introduce two variants of CSB +-Trees. Segmented CSB +-Trees divide the child nodes into segments. Nodes within the same segment are stored contiguously and only pointers to the beginning of each segment are stored explicitly in each node. Segmented CSB +-Trees can reduce the copying cost when there is a split since only one segment needs to be moved. Full

Spatial relations are becoming an important aspect of spatial access methods because of the increasing interest on qualitative spatial information processing. In this paper we show how queries involving spatial relations can be transformed to range queries and implemented in existing DBMSs. We provide a performance analysis of B- and R- tree indexing methods to support such queries and we evaluate the analytical formulas using experimental results. The proposed analytical models for the expected retrieval cost of spatial relations are proved to be good guidelines for a spatial query optimiser.

Previous research has shown that cache behavior is important for main memory index structures. Cache conscious index structures such as Cache Sensitive Search Trees (CSS-Trees) perform lookups much faster than binary search and T-Trees. However, CSS-Trees are designed for decision support workloads with relatively static data. Although B + -Trees are more cache conscious than binary search and T-Trees, their utilization of a cache line is low since half of the space is used to store child pointers. Nevertheless, for applications that require incremental updates, traditional B + -Trees perform well. Our goal is to make B + -Trees as cache conscious as CSS-Trees without increasing their update cost too much. We propose a new indexing technique called "Cache Sensitive B + -Trees" (CSB + -Trees). It is a variant of B + -Trees that stores all the child nodes of any given node contiguously, and keeps only the address of the first child in each node. The rest of the children can ...

With the proliferation of wireless communications and geo-positioning, e-services are envisioned that exploit the positions of a set of continuously moving users to provide context-aware functionality to each individual user. Because advances in disk capacities continue to outperform Moore's Law, it becomes increasingly feasible to store on-line all the position information obtained from the moving e-service users. With the much slower advances in I/O speeds and many concurrent users, indexing techniques are of essence in this scenario. Past

The space utilization of B-tree nodes determines the number of levels in the B-tree and hence its performance. Until now, the only analytical aid to the determination of a B-tree's utilization has been the analysis by Yao and related work. Yao showed that the utilization of B-tree nodes under pure inserts is 69%. We derive analytically and verify by simulation the utilization of B-tree nodes constructed from a mixture of insert and delete operations. Assuming that nodes only merge (i.e. are freed) when they are empty we show that the utilization is 39% when the number of inserts is the same as the number of deletes. However, if there are just 5% more inserts than deletes, then the utilization is over 62%. We also calculate the probability of splitting and merging. We derive a simple rule-of-thumb that accurately calculates the probability of splitting. We also model B-trees that merge half-empty nodes. The utilization of merge-at-half B-trees is slightly larger than the utilization of ...

This article describes a purely analytic approach to urn models of the generalized or extended Pólya-Eggenberger type, in the case of two types of balls and constant "balance", i.e., constant row sum. (Under such models, an urn may contain balls of either of two colours and a fixed 2 × 2-matrix determines the replacement policy when a ball is drawn and its colour is observed.) The treatment starts from a quasilinear first-order partial differential equation associated with a combinatorial renormalization of the model and bases itself on elementary conformal mapping arguments coupled with singularity analysis techniques. Probabilistic consequences are new representations for the probability distribution of the urn's composition at any time n, structural information on the shape of moments of all orders, estimates of the speed of convergence to the Gaussian limits, and an explicit determination of the associated large deviation function. In the general case, analytic solutions involve Abelian integrals over the Fermat curve x = 1. Several urn models, including a classical one associated with balanced trees (2-3 trees and fringe-balanced search trees) and related to a previous study of Panholzer and Prodinger as well as all urns of balance 1 or 2, are shown to admit of explicit representations in terms of Weierstraß elliptic functions. Other consequences include a unification of earlier studies of these models and the detection of stable laws in certain classes of urns with an off-diagonal entry equal to zero.

This thesis develops a validated model of concurrent data structure algorithm performance, concentrating on concurrent B-trees. The thesis first develops two analytical tools, which are explained in the next two paragraphs, for the analysis. Yao showed that the space utilization of a B-tree built from random inserts is 69%. Assuming that nodes merge only when empty, we show that the utilization is 39% when the number of insert and delete operations is the same. However, if there are just 5% more inserts than deletes, then the utilization is at least 62%. In addition to the utilization, we calculate the probabilities of splitting and merging, important parameters for calculating concurrent B-tree algorithm performance. We compare merge-at-empty B-trees with merge-at-half B-trees. We conclude that merge-at-empty Btrees have a slightly lower space utilization but a much lower restructuring rate than merge-at-half B-trees, making merge-at-empty B-trees preferable for concurrent B-tree algo...

Fringe analysis is a technique used to study the average behavior of search trees. In this paper we survey the main results regarding this technique, and we improve a previous asymptotic theorem. At the same time we present new developments and applications of the theory which allow improvements in several bounds on the behavior of search trees. Our examples cover binary search trees, AVL trees, 2-3 trees, and B-trees. Categories and Subject Descriptors: F.2.2 [Analysis of Algorithms and Problem Complexity ]: Nonnumerical Algorithms and Problems -- computations on discrete structures; sorting and searching; E.1 [Data Structures]; trees. Contents 1 Introduction 2 2 The Theory of Fringe Analysis 4 3 Weakly Closed Collections 9 4 Including the Level Information 11 5 Fringe Analysis, Markov Chains, and Urn Processes 13 This work was partially funded by Research Grant FONDECYT 93-0765. e-mail: rbaeza@dcc.uchile.cl 1 Introduction Search trees are one of the most used data structures t...

Aggregate window queries return summarized information about objects that fall inside a query rectangle (e.g., the number of objects instead of their concrete ids). Traditional approaches for processing such queries usually retrieve considerable extra information, thus compromising the processing cost.

Abstract. This paper provides a characterization of the storage needs of a quadtree when used as an index to access large volumes of 2-dimensional data. It is shown that the page occupancy for data in random order approaches 33 %. A precise mathematical analysis that involves a modicum of hypergeometric functions and dilogarithms, together with some computer algebra is presented. A brief survey of the analysis of storage usage in tree structures is included. The 33 % ratio for quadtrees is to be compared to the figures for binary search trees (50%), tries (69%), and quadtries (46%). Computing Reviews Classification: E. 1, E.2, F.2.2, G.2.1. 1. Introduction. The quadtree structure is a fundamental hierarchical representation of point data in higher dimensional spaces. It was invented by Finkel and Bentley in 1974 [7], and it constitutes a natural generalization of binary search trees to multidimensional data. Under one form or other, it has surfaced in many different fields, like data